Catalan conjecture

The Catalan conjecture is a proposition from the mathematical branch of number theory . It is based on the observation that apart from the powers and no other real powers are known that differ by exactly 1. In 1844 Eugène Charles Catalan put forward the Catalan conjecture named after him, according to which there are no other real powers with this property: ${\ displaystyle 2 ^ {3} = 8}$ ${\ displaystyle 3 ^ {2} = 9}$

The only integer solution of the equation with is , , and .

{\ displaystyle x ^ {p} -y ^ {q} = 1}

{\ displaystyle x, p, y, q> 1}

{\ displaystyle x = 3}

{\ displaystyle p = 2}

{\ displaystyle y = 2}

{\ displaystyle q = 3}

Only after 150 years was this assumption proven by Preda Mihăilescu in 2002 .

history

Even before Catalan, related problems were being dealt with. Approx. In 1320 Levi ben Gershon proved : If powers of 2 and 3 differ by 1, then 8 and 9 are the only solutions.

Leonhard Euler (1707–1783) showed that there is only one solution and there is. ${\ displaystyle a ^ {2} -b ^ {3} = 1}$ ${\ displaystyle a = 3}$ ${\ displaystyle b = 2}$

Catalan's conjecture generalizes Euler's equation to general powers. His conjecture was published as a letter to the editor in 1844 in the Journal for Pure and Applied Mathematics .

Later some interesting partial results were found for the case that Catalan's assertion is incorrect, ie that there are other nontrivial solutions to the equation.

In 1976 Robert Tijdeman showed that at most finitely many numbers satisfy the equation.

In 1998 Ray Steiner showed the following property for a possible solution: Either and fulfill certain divisibility conditions ( class number condition ) or and are double Wieferich prime numbers , ie they satisfy the condition ${\ displaystyle p}$ ${\ displaystyle q}$ ${\ displaystyle p}$ ${\ displaystyle q}$

{\ displaystyle p ^ {q-1} \ equiv 1 \ {\ rm {mod}} \ q ^ {2}}

and

{\ displaystyle q ^ {p-1} \ equiv 1 \ {\ rm {mod}} \ p ^ {2}}

In 2000, Maurice Mignotte gave an upper limit for solutions and : q <7.15 × 10 ¹¹ , p <7.78 × 10 ¹⁶ . ${\ displaystyle q}$ ${\ displaystyle p}$

In April 2002, Preda Mihăilescu , who was then employed at the University of Paderborn , finally succeeded in proving the Catalan conjecture, which gave it the status of a mathematical theorem.

generalization

One can extend the now proven Catalan conjecture by using the equation

{\ displaystyle x ^ {p} -y ^ {q} = n}

with natural ,

{\ displaystyle x, y> 0}

{\ displaystyle p, q, n> 1}

considered. It is assumed that this equation, too , only has finitely many solutions for all , so that for every natural number there are only finitely many pairs of real powers whose difference is. ${\ displaystyle x, y> 0}$ ${\ displaystyle p, q, n> 1}$ ${\ displaystyle n \ in \ mathbb {N}}$ ${\ displaystyle n}$

The following list gives up to all the solutions to this equation, which are (for the sake of completeness, this is allowed): ${\ displaystyle n \ leq 64}$ ${\ displaystyle \ leq 10 ^ {18}}$ ${\ displaystyle q = 0}$

List of all the solutions of the equation by using natural , ,

{\ displaystyle x ^ {p} -y ^ {q} = n}

{\ displaystyle 1 \ leq n \ leq 64}

{\ displaystyle 10 ^ {18} \ geq x, y> 0}

{\ displaystyle p, n> 1}

{\ displaystyle q \ geq 0}

n	Number of solutions (sequence A076427 in OEIS )	Numbers such that and both are powers (sequence A103953 in OEIS ) ${\ displaystyle k}$ ${\ displaystyle k + n}$ ${\ displaystyle k}$	${\ displaystyle x ^ {p} -y ^ {q} = n}$
1	1	8th	${\ displaystyle 3 ^ {2} -2 ^ {3} = 1}$
2	1	25th	${\ displaystyle 3 ^ {3} -5 ^ {2} = 2}$
3	2	1, 125	${\ displaystyle 2 ^ {2} -y ^ {0} = 3}$ ${\ displaystyle 2 ^ {7} -5 ^ {3} = 3}$
4th	3	4, 32, 121	${\ displaystyle 2 ^ {3} -2 ^ {2} = 4}$ ${\ displaystyle 6 ^ {2} -2 ^ {5} = 4}$ ${\ displaystyle 5 ^ {3} -11 ^ {2} = 4}$
5	2	4, 27	${\ displaystyle 3 ^ {2} -2 ^ {2} = 5}$ ${\ displaystyle 2 ^ {5} -3 ^ {3} = 5}$
6th	0	there is no solution	-
7th	5	1, 9, 25, 121, 32761	${\ displaystyle 2 ^ {3} -y ^ {0} = 7}$ ${\ displaystyle 4 ^ {2} -3 ^ {2} = 7}$ ${\ displaystyle 2 ^ {5} -5 ^ {2} = 7}$ ${\ displaystyle 2 ^ {7} -11 ^ {2} = 7}$ ${\ displaystyle 32 ^ {3} -181 ^ {2} = 7}$
8th	3	1, 8, 97336	${\ displaystyle 3 ^ {2} -y ^ {0} = 8}$ ${\ displaystyle 2 ^ {4} -2 ^ {3} = 8}$ ${\ displaystyle 312 ^ {2} -46 ^ {3} = 8}$
9	4th	16, 27, 216, 64000	${\ displaystyle 5 ^ {2} -2 ^ {4} = 9}$ ${\ displaystyle 6 ^ {2} -3 ^ {3} = 9}$ ${\ displaystyle 15 ^ {2} -6 ^ {3} = 9}$ ${\ displaystyle 253 ^ {2} -40 ^ {3} = 9}$
10	1	2187	${\ displaystyle 13 ^ {3} -3 ^ {7} = 10}$
11	4th	16, 25, 3125, 3364	${\ displaystyle 3 ^ {3} -2 ^ {4} = 11}$ ${\ displaystyle 6 ^ {2} -5 ^ {2} = 11}$ ${\ displaystyle 56 ^ {2} -5 ^ {5} = 11}$ ${\ displaystyle 15 ^ {3} -58 ^ {2} = 11}$
12	2	4, 2197	${\ displaystyle 2 ^ {4} -2 ^ {2} = 12}$ ${\ displaystyle 47 ^ {2} -13 ^ {3} = 12}$
13	3	36, 243, 4900	${\ displaystyle 7 ^ {2} -6 ^ {2} = 13}$ ${\ displaystyle 2 ^ {8} -3 ^ {5} = 13}$ ${\ displaystyle 17 ^ {3} -70 ^ {2} = 13}$
14th	0	there is no solution	-
15th	3	1, 49, 1295029	${\ displaystyle 2 ^ {4} -y ^ {0} = 15}$ ${\ displaystyle 2 ^ {6} -7 ^ {2} = 15}$ ${\ displaystyle 1138 ^ {2} -109 ^ {3} = 15}$
16	3	9, 16, 128	${\ displaystyle 5 ^ {2} -3 ^ {2} = 16}$ ${\ displaystyle 2 ^ {5} -2 ^ {4} = 16}$ ${\ displaystyle 12 ^ {2} -2 ^ {7} = 16}$
17th	7th	8, 32, 64, 512, 79507, 140608, 143384152904	${\ displaystyle 5 ^ {2} -2 ^ {3} = 17}$ ${\ displaystyle 7 ^ {2} -2 ^ {5} = 17}$ ${\ displaystyle 3 ^ {4} -2 ^ {6} = 17}$ ${\ displaystyle 23 ^ {2} -2 ^ {9} = 17}$ ${\ displaystyle 282 ^ {2} -43 ^ {3} = 17}$ ${\ displaystyle 375 ^ {2} -52 ^ {3} = 17}$ ${\ displaystyle 378661 ^ {2} -5234 ^ {3} = 17}$
18th	3	9, 225, 343	${\ displaystyle 3 ^ {3} -3 ^ {2} = 18}$ ${\ displaystyle 3 ^ {5} -15 ^ {2} = 18}$ ${\ displaystyle 19 ^ {2} -7 ^ {3} = 18}$
19th	5	8, 81, 125, 324, 503284356	${\ displaystyle 3 ^ {3} -2 ^ {3} = 19}$ ${\ displaystyle 10 ^ {2} -3 ^ {4} = 19}$ ${\ displaystyle 12 ^ {2} -5 ^ {3} = 19}$ ${\ displaystyle 7 ^ {3} -18 ^ {2} = 19}$ ${\ displaystyle 55 ^ {5} -22434 ^ {2} = 19}$
20th	2	16, 196	${\ displaystyle 6 ^ {2} -2 ^ {4} = 20}$ ${\ displaystyle 6 ^ {3} -14 ^ {2} = 20}$
21st	2	4, 100	${\ displaystyle 5 ^ {2} -2 ^ {2} = 21}$ ${\ displaystyle 11 ^ {2} -10 ^ {2} = 21}$
22nd	2	27, 2187	${\ displaystyle 7 ^ {2} -3 ^ {3} = 22}$ ${\ displaystyle 47 ^ {2} -3 ^ {7} = 22}$
23	4th	4, 9, 121, 2025	${\ displaystyle 3 ^ {3} -2 ^ {2} = 23}$ ${\ displaystyle 2 ^ {5} -3 ^ {2} = 23}$ ${\ displaystyle 12 ^ {2} -11 ^ {2} = 23}$ ${\ displaystyle 2 ^ {11} -45 ^ {2} = 23}$
24	5	1, 8, 25, 1000, 542939080312	${\ displaystyle 5 ^ {2} -y ^ {0} = 24}$ ${\ displaystyle 2 ^ {5} -2 ^ {3} = 24}$ ${\ displaystyle 7 ^ {2} -5 ^ {2} = 24}$ ${\ displaystyle 2 ^ {10} -10 ^ {3} = 24}$ ${\ displaystyle 736844 ^ {2} -8158 ^ {3} = 24}$
25th	2	100, 144	${\ displaystyle 5 ^ {3} -10 ^ {2} = 25}$ ${\ displaystyle 13 ^ {2} -12 ^ {2} = 25}$
26th	3	1, 42849, 6436343	${\ displaystyle 3 ^ {3} -y ^ {0} = 26}$ ${\ displaystyle 35 ^ {3} -207 ^ {2} = 26}$ ${\ displaystyle 2537 ^ {2} -23 ^ {5} = 26}$
27	3	9, 169, 216	${\ displaystyle 6 ^ {2} -3 ^ {2} = 27}$ ${\ displaystyle 14 ^ {2} -13 ^ {2} = 27}$ ${\ displaystyle 3 ^ {5} -6 ^ {3} = 27}$
28	7th	4, 8, 36, 100, 484, 50625, 131044	${\ displaystyle 2 ^ {5} -2 ^ {2} = 28}$ ${\ displaystyle 6 ^ {2} -2 ^ {3} = 28}$ ${\ displaystyle 2 ^ {6} -6 ^ {2} = 28}$ ${\ displaystyle 2 ^ {7} -10 ^ {2} = 28}$ ${\ displaystyle 2 ^ {9} -22 ^ {2} = 28}$ ${\ displaystyle 37 ^ {3} -225 ^ {2} = 28}$ ${\ displaystyle 2 ^ {17} -362 ^ {2} = 28}$
29	1	196	${\ displaystyle 15 ^ {2} -14 ^ {2} = 29}$
30th	1	6859	${\ displaystyle 83 ^ {2} -19 ^ {3} = 30}$
31	2	1, 225	${\ displaystyle 2 ^ {5} -y ^ {0} = 31}$ ${\ displaystyle 16 ^ {2} -15 ^ {2} = 31}$
32	4th	4, 32, 49, 7744	${\ displaystyle 6 ^ {2} -2 ^ {2} = 32}$ ${\ displaystyle 2 ^ {6} -2 ^ {5} = 32}$ ${\ displaystyle 3 ^ {4} -7 ^ {2} = 32}$ ${\ displaystyle 6 ^ {5} -88 ^ {2} = 32}$

n	Number of solutions (sequence A076427 in OEIS )	Numbers such that and both are powers (sequence A103953 in OEIS ) ${\ displaystyle k}$ ${\ displaystyle k + n}$ ${\ displaystyle k}$	${\ displaystyle x ^ {p} -y ^ {q} = n}$
33	2	16, 256	${\ displaystyle 7 ^ {2} -2 ^ {4} = 33}$ ${\ displaystyle 17 ^ {2} -2 ^ {8} = 33}$
34	0	there is no solution	-
35	3	1, 289, 1296	${\ displaystyle 6 ^ {2} -y ^ {0} = 35}$ ${\ displaystyle 18 ^ {2} -17 ^ {2} = 35}$ ${\ displaystyle 11 ^ {3} -36 ^ {2} = 35}$
36	2	64, 1728	${\ displaystyle 10 ^ {2} -2 ^ {6} = 36}$ ${\ displaystyle 42 ^ {2} -12 ^ {3} = 36}$
37	3	27, 324, 14348907	${\ displaystyle 2 ^ {6} -3 ^ {3} = 37}$ ${\ displaystyle 19 ^ {2} -18 ^ {2} = 37}$ ${\ displaystyle 3788 ^ {2} -243 ^ {3} = 37}$
38	1	1331	${\ displaystyle 37 ^ {2} -11 ^ {3} = 38}$
39	4th	25, 361, 961, 10609	${\ displaystyle 2 ^ {6} -5 ^ {2} = 39}$ ${\ displaystyle 20 ^ {2} -19 ^ {2} = 39}$ ${\ displaystyle 10 ^ {3} -31 ^ {2} = 39}$ ${\ displaystyle 22 ^ {3} -103 ^ {2} = 39}$
40	4th	9, 81, 216, 2704	${\ displaystyle 7 ^ {2} -3 ^ {2} = 40}$ ${\ displaystyle 11 ^ {2} -3 ^ {4} = 40}$ ${\ displaystyle 2 ^ {8} -6 ^ {3} = 40}$ ${\ displaystyle 14 ^ {3} -52 ^ {2} = 40}$
41	3	8, 128, 400	${\ displaystyle 7 ^ {2} -2 ^ {3} = 41}$ ${\ displaystyle 13 ^ {2} -2 ^ {7} = 41}$ ${\ displaystyle 21 ^ {2} -20 ^ {2} = 41}$
42	0	there is no solution	-
43	1	441	${\ displaystyle 22 ^ {2} -21 ^ {2} = 43}$
44	3	81, 100, 125	${\ displaystyle 5 ^ {3} -3 ^ {4} = 44}$ ${\ displaystyle 12 ^ {2} -10 ^ {2} = 44}$ ${\ displaystyle 13 ^ {2} -5 ^ {3} = 44}$
45	4th	4, 36, 484, 9216	${\ displaystyle 7 ^ {2} -2 ^ {2} = 45}$ ${\ displaystyle 9 ^ {2} -6 ^ {2} = 45}$ ${\ displaystyle 23 ^ {2} -22 ^ {2} = 45}$ ${\ displaystyle 21 ^ {3} -96 ^ {2} = 45}$
46	1	243	${\ displaystyle 17 ^ {2} -3 ^ {5} = 46}$
47	6th	81, 169, 196, 529, 1681, 250000	${\ displaystyle 2 ^ {7} -3 ^ {4} = 47}$ ${\ displaystyle 6 ^ {3} -13 ^ {2} = 47}$ ${\ displaystyle 3 ^ {5} -14 ^ {2} = 47}$ ${\ displaystyle 24 ^ {2} -23 ^ {2} = 47}$ ${\ displaystyle 12 ^ {3} -41 ^ {2} = 47}$ ${\ displaystyle 63 ^ {3} -500 ^ {2} = 47}$
48	4th	1, 16, 121, 21904	${\ displaystyle 7 ^ {2} -y ^ {0} = 48}$ ${\ displaystyle 2 ^ {6} -2 ^ {4} = 48}$ ${\ displaystyle 13 ^ {2} -11 ^ {2} = 48}$ ${\ displaystyle 28 ^ {3} -148 ^ {2} = 48}$
49	3	32, 576, 274576	${\ displaystyle 3 ^ {4} -2 ^ {5} = 49}$ ${\ displaystyle 25 ^ {2} -24 ^ {2} = 49}$ ${\ displaystyle 65 ^ {3} -524 ^ {2} = 49}$
50	0	there is no solution	-
51	2	49, 625	${\ displaystyle 10 ^ {2} -7 ^ {2} = 51}$ ${\ displaystyle 26 ^ {2} -5 ^ {4} = 51}$
52	1	144	${\ displaystyle 14 ^ {2} -12 ^ {2} = 52}$
53	2	676, 24336	${\ displaystyle 27 ^ {2} -26 ^ {2} = 53}$ ${\ displaystyle 29 ^ {3} -156 ^ {2} = 53}$
54	2	27, 289	${\ displaystyle 3 ^ {4} -3 ^ {3} = 54}$ ${\ displaystyle 7 ^ {3} -17 ^ {2} = 54}$
55	3	9, 729, 175561	${\ displaystyle 2 ^ {6} -3 ^ {2} = 55}$ ${\ displaystyle 28 ^ {2} -27 ^ {2} = 55}$ ${\ displaystyle 56 ^ {3} -419 ^ {2} = 55}$
56	4th	8, 25, 169, 5776	${\ displaystyle 2 ^ {6} -2 ^ {3} = 56}$ ${\ displaystyle 3 ^ {4} -5 ^ {2} = 56}$ ${\ displaystyle 15 ^ {2} -13 ^ {2} = 56}$ ${\ displaystyle 18 ^ {3} -76 ^ {2} = 56}$
57	3	64, 343, 784	${\ displaystyle 11 ^ {2} -2 ^ {6} = 57}$ ${\ displaystyle 20 ^ {2} -7 ^ {3} = 57}$ ${\ displaystyle 29 ^ {2} -28 ^ {2} = 57}$
58	0	there is no solution	-
59	1	841	${\ displaystyle 30 ^ {2} -29 ^ {2} = 59}$
60	4th	4, 196, 2515396, 2535525316	${\ displaystyle 2 ^ {6} -2 ^ {2} = 60}$ ${\ displaystyle 2 ^ {8} -14 ^ {2} = 60}$ ${\ displaystyle 136 ^ {3} -1586 ^ {2} = 60}$ ${\ displaystyle 76 ^ {5} -50354 ^ {2} = 60}$
61	2	64,900	${\ displaystyle 5 ^ {3} -2 ^ {6} = 61}$ ${\ displaystyle 31 ^ {2} -30 ^ {2} = 61}$
62	0	there is no solution	-
63	4th	1, 81, 961, 183250369	${\ displaystyle 2 ^ {6} -y ^ {0} = 63}$ ${\ displaystyle 12 ^ {2} -9 ^ {2} = 63}$ ${\ displaystyle 2 ^ {10} -31 ^ {2} = 63}$ ${\ displaystyle 568 ^ {3} -13537 ^ {2} = 63}$
64	4th	36, 64, 225, 512	${\ displaystyle 10 ^ {2} -6 ^ {2} = 64}$ ${\ displaystyle 2 ^ {7} -2 ^ {6} = 64}$ ${\ displaystyle 17 ^ {2} -15 ^ {2} = 64}$ ${\ displaystyle 24 ^ {2} -2 ^ {9} = 64}$

literature

Preda Mihailescu: Primary cyclotomic units and a proof of Catalan's conjecture. J. Reine Angew. Math. 572 (2004), 167--195
Christoph Pöppe: The proof of the Catalan conjecture. In: Omega. The magazine for math, logic and computers. ( Spectrum of Science Special 4/2003) Spektrumverlag, Heidelberg 2003, pp. 64–67
Yuri Bilu: Catalan's Conjecture (after Mihailescu). Seminaire Bourbaki, No. 909, 2002, ( PDF ).
Jeanine Daems: A Cyclotomic Proof of Catalan's Conjecture. Diploma thesis, University of Leiden 2003, ( PDF ).
Maurice Mischler, Jacques Boéchat on the Catalan Assumption, French ( Arxiv ).
Henri Cohen on the proof of the Catalan Conjecture, French ( online ).

Web links

Eric Weisstein: Catalan's Conjecture . In: MathWorld (English).
https://www.welt.de/print-wams/article604626/Geniestreich-eines-spaet-Berufenen.html

Individual evidence

^ Eugène Charles Catalan : Note extraite d'une lettre adressée à l'éditeur par Mr. E. Catalan, Répétiteur à l'école polytechnique de Paris. Journal for pure and applied mathematics 27, 192, 1844 (scan of the original online) (accessed April 16, 2019)

[CATALAN-1] Eugène Charles Catalan : Note extraite d'une lettre adressée à l'éditeur par Mr. E. Catalan, Répétiteur à l'école polytechnique de Paris. Journal for pure and applied mathematics 27, 192, 1844 (scan of the original online) (accessed April 16, 2019)